Built-in self test of MEMS

ABSTRACT

The present disclosure is directed to an apparatus and method for producing and comparing signals from various points in a MEMS device. By producing signals which should be of substantial identical characteristics, deviations from the situation where the signals are of identical characteristics can be used to identify various types of asymmetry which are otherwise difficult to detect. In one embodiment, the MEMS device is comprised of a plurality of fixed beams arranged symmetrically and a plurality of movable beams arranged symmetrically. A first sensor is formed by certain of the fixed and movable beams while a second sensor, electrically isolated from said first sensor, is formed by at least certain other of the fixed and movable beams. The first and second sensors are located within the MEMS device so as to produce signals of substantially identical characteristics. A circuit is responsive to the first and second sensors for comparing the signals produced by the first and second sensors. In addition to the apparatus, methods of performing a self test are also disclosed, which may be performed in real time.

CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority from provisional applicationserial No. 60/411,703 filed Sep. 18, 2003 and entitled Built-In SelfTest of CMOS-MEMS Accelerometers, the entirety of which is herebyincorporated by reference.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

[0002] This work was sponsored by the National Science Foundation undergrant MIP-9702678. The federal government may have rights in thisinvention.

BACKGROUND

[0003] The present disclosure is directed to a built-in, self-testtechnique for MicroElectroMechanical systems (MEMS) that is applicableto symmetrical microstructures or other structures in which signals ofequal magnitude and opposite polarity can be produced.

[0004] MEMS are complex, heterogeneous systems consisting of deviceswhose operation is based on the interactions of multiple energy domains.Commercial manufacture of MEMS has increased the need for cost-effectivetest methods that screen defective devices from good ones. With MEMSbecoming increasingly complex and finding use in life-criticalapplications such as air-bags, bio-sensors, and aerospace applications,there is a growing need for robust fault models and test methods.

[0005] Of the currently used MEMS process technologies, surfacemicromachining is a popular one due to its well-developed infrastructurefor depositing, patterning and etching of thin films for siliconintegrated circuits. Surface micromachining enables the fabrication ofhigh-quality MEMS devices because it is based on thin-film technologythat combines control and flexibility in the fabrication process.

[0006] A MEMS test may include the process of identifying good devicesin a batch of fabricated devices. The normal assumption is that thedesign is correct so the test process is one of verifying that thefabricated device is equivalent to the design. However, a device thatpasses a traditional, specification-based test may fail later duringin-field operation. For example, a mechanical beam of an accelerometermay become stuck to the die surface due to a phenomenon known asstiction. A stuck beam may mimic behavior similar to a device affectedby an expected level of under-etch. Under-etch refers to a release stepwhere sacrificial material is removed to release or free themicrostructure. The release typically requires an etching step, and inthe case where an under-etch occurs, insufficient sacrificial materialmay be removed. By adjusting the electronics, an accelerometer sufferingfrom stiction can be easily calibrated to meet its operationalspecification. The danger, however, is that an accelerometer with astuck beam may release in the field (i.e. defect healing) causing theaccelerometer to go out of calibration, which can then possibly lead tofailure. Detection or prevention of field failures can be accomplishedthrough built-in self test (BIST).

[0007] As MEMS become more complicated and find a wider range ofapplications, the need for on-chip self-test features will grow. BISTfor the testing of MEMS is yet to be common practice. However, progressin this area has been recently made. The work in De Bruyker et al., “ACombined Piezoresistive/Capacitive Pressure Sensor with Self-testFunction based on Thermal Actuation,” Proc. Solid State Sensors andActuators, Vol. 2, pp. 1461-1464 (1997) describes the self-test of apressure sensor. In their approach, thermal actuation of the sensor'sdiaphragm is performed by driving current through a resistive heater.The heat generated increases the temperature of the air in the sensor'scavity creating a pressure that displaces the diaphragm. Using a similartechnique, the authors Charlot et al., of “Electrically Induced Stimulifor MEMS Self-Test,” Proc. VLSI Test Symposium, pp. 210-215 (Apr.-May2001) use resistive heaters to increase the temperature of a MEMSinfrared-imager array. Many commercial accelerometers use a self-testtechnique similar to the one described in Allen et al., “Self-TestableAccelerometer Systems,” Proc. Micro Electro Mechanical Systems, pp.113-115 (1989). In that approach, dedicated mechanical beams are used togenerate an electrostatic force that mimics an external acceleration. Itis useful for determining if the accelerometer's mechanicalmicrostructure is free to move. This technique cannot be used until theelectrostatic force is calibrated after testing has been performed todetermine if the part has been manufactured correctly. Finally, an ideafor accelerometer self-test that exploits design symmetry is suggestedin Rosing et al., “Fault Simulation and Modeling ofMicroelectromechanical Systems,” Computing and Control EngineeringJournal, Vol. 11, Issue 5, pp. 242-250 (October 2000). They propose toactuate the accelerometer one side at a time and then compare the twooutputs obtained to detect any anomaly.

[0008] Commercially-manufactured devices such as accelerometers areusually affected by multiple failure sources. Failure sources for MEMSinclude, but are not limited to, foreign particles, etch variations, andstiction, each of which can lead to a variety of defects. For example,it is known that particles can lead to defects that include broken andbridged structures with corresponding behaviors that range betweenbenign and catastrophic. Many of these failure sources exhibit verysimilar misbehaviors and are difficult to distinguish from each other.

[0009] Currently, self-test of commercial accelerometers is limited.BIST techniques used in industry (See Allen et al., “Self-TestableAccelerometer Systems,” Proc. Micro Electro Mechanical Systems, pp.113-115 (1989)) are focused on input stimulus generation. Inaccelerometers produced by Analog Devices, Motorola and others, theaccelerometer's shuttle is moved to its maximum position using actuationfingers so that the full-scale sense output is generated. The inabilityto generate a full-scale output, within some tolerance limits, means theaccelerometer has failed self test. Using this form of BIST for testingis difficult because the amount of actuation voltage needed can bedetermined only after the part has been tested and calibrated. It isalso ineffective for distinguishing misbehavior stemming from differentsources. For example, a BIST output that is larger (smaller) thanexpected can be either caused by over-etch (under-etch) or broken(stuck) beams. Hence, its ability to identify hard-to-detect defects(e.g., asymmetry due to local defects) is limited.

[0010] In the BIST technique proposed in Rosing et al., supra, theaccelerometer's shuttle is moved twice, once using the right actuationfingers and again using the left actuation fingers. Failure results whenthe two resulting sense outputs do not match, presumably, within sometolerance level. Unlike the techniques currently used in industry, thismethod does not necessarily require calibration before it can be used.However, its ability to identify hard-to-detect defects is limitedbecause all test observations are made from the normal sense output.Moreover, it is difficult to implement on chip because sample-and-holdcircuitry is required to store the first measurement.

SUMMARY

[0011] According to one embodiment of the disclosure, the improvement ina symmetric MEMS device comprises first and second sensors electricallyisolated from one another and positioned to produce signals ofsubstantially identical characteristics (e.g, producing signals of equalmagnitude but opposite polarity). A circuit is responsive to the sensorsfor comparing the signals produced by the sensors. The comparison may beperformed in real time.

[0012] According to another embodiment of the disclosure, a MEMS devicecomprises a plurality of fixed beams arranged symmetrically and aplurality of movable beams arranged symmetrically. A first sensor isformed by certain of the fixed and movable beams while a second sensoris formed by at least certain other of the fixed and movable beams. Thefirst and second sensors are symmetrically located within the MEMSdevice. A circuit is responsive to the first and second sensors forcomparing signals produced by the first and second sensors. Thecomparison may be performed in real time.

[0013] The present disclosure is also directed to a method comprisingactuating a MEMS device and comparing the outputs from a first and asecond sensor, each electrically isolated from one another andpositioned to produce signals of substantially identicalcharacteristics. The sensors may be symmetrically located and the methodmay be carried out in real time.

[0014] A combination of existing layout features and additionalcircuitry is used to make measurements from various points in a MEMS. Inaddition to the normal sense output, self-test outputs are used todetect the presence of layout asymmetry that are caused by local,hard-to-detect defects. Simulation results for an accelerometer revealthat the disclosed approach is able to distinguish misbehavior resultingfrom local defects and manufacturing process variations.

BRIEF DESCRIPTION OF THE DRAWINGS

[0015] For the present disclosure to be easily understood and readilypractice, the disclosure will now be described, for purposes ofillustration and not limitation, in conjunction with the followingfigures where:

[0016] FIGS. 1A-1D illustrate various examples of symmetric MEMS devicessuch as (A) microresonator, (B) accelerometer, (C) gyroscope, (D) arrayof RF-MEMS switches in an antenna, (E) micromotor, and (D) ink-jet printhead microsystem with 50 nozzles;

[0017]FIG. 2 is a simplified top view of an accelerometer's mechanicalmicrostructure;

[0018]FIG. 3 is a table illustrating microstructure voltage biasing fornormal and self-test operations;

[0019]FIG. 4 is a schematic of a differential amplifier;

[0020] FIGS. 5A-5C are schematics showing the modulation schemes for (A)fully differential normal operation, (B) differential BIST operationusing normal sense fingers and (C) differential BIST operation usingspring beams;

[0021]FIG. 6 illustrates the topology of a test accelerometer showingthe serpentine springs;

[0022]FIG. 7 illustrates sets of comb-drives, each showing a pair ofcapacitors and two free nodes;

[0023]FIG. 8 illustrates the layout of the BIST design showing thelocation of the sensors and the switch-based BIST control; and

[0024]FIG. 9 is a schematic illustrating one example of controlcircuitry for use with BIST.

DESCRIPTION

[0025] This disclosure describes a BIST approach that samples outputsfrom symmetrically-located nodes of a MEMS microstructure. Increasingobservability in this way allows one to identify misbehavior resultingfrom local defects as opposed to more benign causes. The disclosed BISTapproach builds upon the fully differential sensing technique describedin Luo et al., “A 1 mG Lateral CMOS-MEMS Accelerometer,” Proc. of MicroElectro Mechanical Systems, pp 502-507 (January 2000) and applies to abroad class of sensors and actuators that includes resonators,accelerometers, and gyroscopes. Examples of symmetric MEMS devices thatfall within the scope of the disclosed BIST approach are illustrated inFIG. 1. We have focused on CMOS-MEMS because the availability ofmultiple routing layers makes BIST in CMOS-MEMS more easily implementedthan in technologies where the routing of wires is limited; other typesof batch-fabrication techniques may be used.

[0026] In our previous work we have shown that changes in devicebehavior due to global manufacturing variations (such asover/under-etch) may mimic those caused by point defects (such asparticles). In such cases of misbehavior overlap, distinguishing betweenvarious failure sources becomes difficult. Among failure sourcesexhibiting similar misbehavior, the potential long-term effects of someare expected to be more harmful than others. Because misbehavior overlaphampers defect diagnosis, it also prevents more harmful defects frombeing distinguished from those which are benign. The BIST approachdisclosed here resolves that issue through differential actuation andsensing. Our method uses existing device features to produce two signalsthat should be identical in the nominal design but are unequal whenasymmetry exists. The disclosed BIST allows one to distinguish betweenharmful defects that cause asymmetries as opposed to those that preservelayout symmetry. Structural deformations that preserve layout symmetryare usually caused by normal process variations. Because those changesare permanent, it is safe to compensate for those variations throughelectronic calibration.

[0027] The BIST of the present disclosure is explained in the context ofa CMOS-MEMS accelerometer for purposes of explanation and notlimitation. A simplified view of an accelerometer's mechanicalmicrostructure is shown in FIG. 2. For purposes of clarity, we haveomitted details of the serpentine spring structure, the multi-layereddevice structure, routed interconnects, and the like.

[0028] A MEMS accelerometer is a transducer that converts translationalacceleration to an electrical signal that is typically a voltage. Anaccelerometer's mechanical component (i.e., the sensor) can be viewed asa collection of primitive microstructures that include beams, anchorsand a plate called the shuttle. Anchors attach beams to the die surfaceonly at the positions shown in FIG. 2. Anchored beams connected to theshuttle act as springs because they create a restoring force when theshuttle moves as a result of an input acceleration. The remaining beamsare typically referred to as “fingers”.

[0029] Accelerometer fingers are partitioned into two classes: fixed andmovable. Fixed fingers are anchored to the die surface and therefore arenot free to move. Movable fingers are attached to the shuttle andtherefore can move along with the shuttle. Subsets of fixed and movablefingers also serve various purposes. The sense fingers enablemeasurement of shuttle movement in the X direction while actuationfingers are used to create an electrostatic force that moves the shuttlefor testing purposes. Dummy fingers are not involved in the normaloperation of the accelerometer but are used to enhance themanufacturability of the device.

[0030] An accelerometer's sensor behaves as a linear second-order systemsimilar to a spring-mass-damper system. As already mentioned, the beamsattached to the top and bottom of the shuttle act as restoring springs.The shuttle is capable of motion by virtue of the flexibility providedby these so-called spring beams. Motion at or near the anchor points isnegligible so locations farthest away from the anchors experience thegreatest amount of movement. Damping of the accelerometer is caused bythe air surrounding it.

[0031] In response to an input acceleration, the shuttle moves from itsresting position until the restoring force of the spring beams balancesthe inertial force caused by the acceleration. Each triplet offixed-movable-fixed fingers constitutes a pair of capacitors, C₁ and C₂,as shown in FIG. 2. At rest, the two capacitors are equal. Shuttlemovement however causes the value of one capacitor to increase and theother to decrease. Shuttle movement is detected or “sensed” byelectronics that detects change in the capacitances. With modulationvoltage signals (e.g., high frequency pulse train) of opposite phasesapplied to the fixed fingers, the finger triplet is a potential dividerwith the voltage output of the movable finger serving as the outputsense signal. In a fully-differential sensing scheme, one phase of themodulation voltage (V_(mp)) is applied to the finger pairs [S₁S₃] and[S₅S₇] and the other phase (V_(mn)) is applied to [S₂S₄] and [S₆S₈] (seeFIG. 5A and FIG. 2). A sense signal from electrically-connected sensefingers on the left (M₅ and M₇) and right (M₆ and M₈) sides areconnected to inputs A₁ and A₂ of a differential amplifier (FIG. 4),respectively, where the primary sense outputs V_(sp) V_(s1) V_(s2) andV_(sn) V_(s2) V_(S1) are produced. The fully-differential scheme ofsensing has the advantage of rejecting any noise that is common to theleft and right sides of the sensor.

[0032] An electrostatic attraction force can be created to displace theshuttle for testing. The voltage of the movable fingers for a shuttle atrest is called the nominal voltage (Vnom), and is simply the average ofthe modulation signal, namely$\frac{1}{T}{\int_{0}^{T}{V_{m\quad p}\quad {{t}.}}}$

[0033] Electrostatic actuation in the positive X direction is achievedby applying the actuation signals (Vact), usually DC or low frequency,to fingers [F1,F3,F5,F7] and the nominal voltage to the fingers[F2,F4,F6,F8], [M1-M4], and [D1-D8] (see FIG. 2).

[0034] The differential BIST approach disclosed herein detectshard-to-detect defects that occur during manufacture or operation in thefield. Its differential nature implies that it is independent of anycalibration and therefore is also suitable for testing purposes. Thedisclosed BIST allows one to distinguish between defects that lead toasymmetries as opposed to those more benign deformations that preservelayout symmetry. The disclosed BIST is primarily targeted at defectescapes, that is, those MEMS parts that would be viewed as good partsdue to the inability of current test practices to identify them asdefective. The self-test technique is focused on observation andtherefore complements existing and proposed approaches that focus onstimulus generation. It creates and compares signals from pairs ofsymmetrically-located points on the accelerometer's micromechanicalstructure. Specifically, these sense points include the normal sensefingers and the spring beams that are surrounded by dummy fingers. Dummyfingers are manufacturing-enhancing structures located near the springbeams (see FIG. 2). They are used to ensure that the spring beams haveetch-loading properties similar to those experienced by the movable andfixed fingers. During self-test, the dummy fingers may have additionalmodulation signals applied and the spring beams are used as additionalsensing signals. The differential amplifier of FIG. 4 is used to detectany difference between symmetrically-located fingers and beams. Thesignal nodes that are compared must all be electrically isolated fromeach other, a feature that is easily achieved using the multiple routinglayers available in a CMOS process. For each sense-point pair, aseparate differential amplifier is used. Additional amplifiers forself-test increases cost in terms of area overhead but has the advantageof reduced parasitic capacitance and interference. Local defects thatintroduce undesirable asymmetry between the left and right sides willcause the two symmetrically located sense outputs, V_(s1) and V_(s2), tobe unequal. If the difference |V_(s1)−V_(s2)|>T, where T is somepre-determined threshold, then asymmetry is detected. The polarity ofthe difference signal V_(s1)-V_(s2) also grossly localizes the defectsite. For example, a contamination that creates a high-resistive bridgebetween fingers F₁ and M₁ will hinder the motion of the sensor's leftside more than that of the right side. Hence, the right sense outputV_(s1)(from [M₅,M₇]) will be less than V_(s2) (from [M₆,M₈]) causingV_(s1)-V_(s2) to be negative. In the opposite case, V_(s1)-V_(s2) willbe positive.

[0035] The operational details of one embodiment of the self-test schemeare now explained. To move the shuttle in the positive X direction, thenominal voltage is applied to fingers [F₂F₄F₆F₈], [M₁ M₄], and [D₅D₈]and actuation voltages are applied to fingers [F₁F₃F₅F₇] (see FIG. 2 andFIG. 3). The sense signal is sampled from different pairs of points.Sensing from the regular sense fingers for self-test purposes can beeasily achieved by the scheme illustrated in FIG. 5B. Modulation signalsof one phase (V_(mp)) are applied to the finger pairs [S₁S₄] and [S₅S₈]while those of the other phase (V_(mn)) are applied to the finger pairs[S₂S₃] and [S₆S₇]. Note that this scheme is derived from the nominal oneshown in FIG. 5A by interchanging the phases of the modulation signalson the right side. Sense signals from finger pairs [M₅M₇] and [M₆M₈] aredirected to the inputs A₁ and A₂, respectively, of a dedicateddifferential amplifier like the one shown in FIG. 4.

[0036] For sensing from the spring beams, opposite phases of modulationsignals are applied to dummy finger pairs [D₁D₃] and [D₂D₄] (see FIG.5C). The sense signals from spring beams B₁ and B₂ are directed to theinputs A₁ and A₂, respectively, of another dedicated differentialamplifier. The use of dedicated amplifiers, and sensors electricallyisolated from one another, enables real time comparison and analysis ifdesired.

[0037] Self-test using movement in the negative X direction can beachieved in a similar fashion. The multi-conductor nature of CMOS-MEMSallows extensive routing and electrical isolation of fingers that aremechanically connected to the same shuttle, characteristics that makeimplementation of this BIST approach practicable. For example, inCMOS-MEMS, the sense voltage outputs from the spring beams can beelectrically isolated very easily. This type of electrical isolation ismuch harder to achieve in other processes such as MUMPS because of itssingle conducting layer.

[0038] The accelerometer is susceptible to both global and localmanufacturing variations. Because the disclosed BIST technique detectsleft-right asymmetry, it is clear that global variations will affectboth sides equally and therefore V_(s1) and V_(s2) will be affectedequally, implying V_(s1) V_(s2) will, in theory, be unaffected. Butlocal manufacturing variations (such as local over/under etch, curvaturevariations, etc.) that lead to left-right asymmetry will result inunequal actuation forces on the left and right sides which in turn willlead to unequal sense outputs. Hence, this self-test approach issensitive to local variations while being immune to global variations.For this reason, the difference V_(s1) V_(s2) or the ratio V_(s1) V_(s2)is more significant than the absolute values of V_(s1) and V_(s2).

[0039] Ideally, the self-test mechanism itself must not falsely indicatedevice failure. So the self-test actuation scheme should prevent thesense signal from being corrupted by the actuation signal even in thepresence of left-right asymmetry caused by local manufacturingvariations. The differential nature of the disclosed process guaranteesthat the actuation signal (i.e., voltage) of each polarity is equallyapplied to both the left and right sides. In short, one polarity isapplied to the fingers [F₁F₇] and the other to [F₃F₅]. If, for example,the inter-finger capacitors on the right side are smaller than those onthe left side, the resulting currents will therefore be unequal (less onthe right side). The differential nature of the disclosed approachguarantees that current due to the actuation signal of each polarity hascontributions from both the left (higher capacitance) and right (lowercapacitance) sides. Therefore, the total current of each polarity is thesame. Hence, given first-order manufacturing variations, the currents ofthe two polarities cancel and do not contribute to the differentialself-test sense signal.

[0040] Analysis

[0041] In the following, we formally show how the disclosed BIST forsymmetric MEMS is able to detect and identify various forms of asymmetryintroduced by perturbations in the manufacturing process. Specifically,the analysis is applied to a MEMS accelerometer; however similaranalyses can be applied to other types of symmetric MEMS.

[0042] We first analyze how the sense comb capacitances, under differentmodulation schemes, interact to produce a sense signal that may or maynot depend on first-order local manufacturing variations. Note from FIG.6 that the accelerometer (used as an example of a symmetric MEMSstructure) has four identical, symmetrically-located sense comb-drives.The network constituted by the capacitances in the four sensecomb-drives is shown in FIG. 7. The symbols ‘l’, ‘r’, ‘b’, and ‘t’denote left, right, bottom and top, respectively. The symbols ‘1’ and‘2’ denote the two capacitors of a differential pair. For shuttle motionin the +Y direction, all capacitors with a suffix of ‘1’ decrease, whileall capacitors with a suffix of ‘2’ increase. The opposite holds truefor motion in the −Y direction.

[0043] The following assumptions are true for the comb-capacitancesdisplayed in FIG. 7.

[0044] 1. Each pair of capacitors constitutes a potential dividercircuit with the potentials applied at the free node (modulation node)of each capacitor and voltage sensed from their common node (sensenode). Thus, each pair has one sense node and two modulation nodes.

[0045] 2. Each pair must have different voltages applied at theirmodulation nodes, for the sense node output to be sensitive to change inthe capacitances. Usually the modulation signals are applied in oppositephases. Therefore each capacitor pair can be considered a dipole.

[0046] 3. The number of freedoms associated with each dipole is m(m−1),where m is the number of phases of the applied modulation signal. Hence,a dipole can be in any one of the m(m−1) possible modulation states.

[0047] Based on the above assumptions, we can find out the number ofmodulation schemes possible for a capacitive network.

[0048] 1. In our case, each dipole can have two possible states,represented by the symbols 1 or 0. Every dipole has two capacitors, onewith ‘1’ in the suffix and the other with ‘2’ in the suffix. If thepositive phase (V_(mp)) of the modulation signal is applied to the freenode of the capacitor with suffix ‘1’, then the negative phase (V_(mn))of the modulation signal, naturally, is applied to the free node of thecapacitor with suffix ‘2’, and the dipole is said to be in state 0.

[0049] 2. A capacitive network consisting of n dipoles can be in one of2^(n)′ states. However, exchange of the two modulation signals for allthe dipoles will change nothing (except polarity of sense output).Hence, only 2^(n−1) states need to be considered. In our case, n=4,which implies only 8 states of the network need to be analyzed. Thisnumber can be further reduced by the heuristic that only states thathave equal number of 1's and 0's are interesting, besides the all-zerostate (or the all-one state). This translates to a value of 4. TABLE 1Definitions of symbols. Symbol Definition C₀ Nominal value of a sensecapacitor of a dipole, in absense of any local and global manufacturingvariations δ Nominal value of the change in a sense capacitor of adipole, in absense of any local and global manufacturing variations, fora specified displacement. Note that |δ/C₀| ≦0.05. C_(p) Total parasiticcapacitance between a sense node and ground. It includes amplifier inputcapacitance, inter-connect capacitance, and all other capacitorsconnected (from non- sensing nodes) to the sense node. C_(total) Totalcapacitance between a sense node and ground. V_(A) Sense voltage outputat node A V_(B) Sense voltage output at node B S_(i) A binary variableassociated with dipole i. Its value is = 1, if dipole i is in state 0 =−1, if dipole i is in state 1 λ_(i) Effective local manufacturingvariation factor for the capacitances of dipole i. Its value is unity inabsence of any manufacturing variations. It is assumed to be constantfor small (typical) displacements of the movable parts of the device.

[0050] If we impose the condition that the sum of the capacitors in adipole is constant (for small displacements), we can obtain (using thesymbol definitions in Table 1) the unified expression for all modes as:$\begin{matrix}{\frac{{\Delta \quad V_{B}} - {\Delta \quad V_{A}}}{V_{m\quad p} - V_{m\quad n}} = {\frac{{S_{I\quad b}\Delta \quad C_{{1l\quad b}\quad}} + {S_{r\quad t}\Delta \quad C_{1\quad r\quad t}}}{\left( {C_{1l\quad b} + C_{2l\quad b}} \right) + \left( {C_{1\quad r\quad t} + C_{2\quad r\quad t}} \right) + C_{p_{B}}} - \frac{{S_{I\quad t}\Delta \quad C_{1\quad l\quad t}} + {S_{rb}\Delta \quad C_{1\quad r\quad b}}}{\left( {C_{1\quad r\quad t} + C_{2\quad r\quad t}} \right) + \left( {C_{1\quad r\quad b} + C_{2\quad {rb}}} \right) + C_{p_{A}}}}} & (1)\end{matrix}$

[0051] Table 2 lists the values of comb capacitances in the presence ofmanufacturing variations. Based on the symbol definitions in Table 1 andthe values listed in Table 2, we have $\begin{matrix}{{{\Delta \quad C_{1{lt}}} = {{- \delta}\quad \lambda_{lt}}},{{\Delta \quad C_{1{rb}}} = {{- \delta}\quad \lambda_{rb}}},{{\Delta \quad C_{1l\quad b}} = {{- \delta}\quad \lambda_{l\quad b}}},} \\\begin{matrix}{{{\Delta \quad C_{1{rt}}} = {{- \delta}\quad \lambda_{rt}}},C_{A_{total}}} \\{{= {{2{C_{0}\left( {\lambda_{lt} + \lambda_{rb}} \right)}} + C_{p_{A}}}},C_{B_{total}}} \\{= {{2{C_{0}\left( {\lambda_{lt} + \lambda_{rb}} \right)}} + {C_{P_{B}}.}}}\end{matrix}\end{matrix}$

[0052] Using these values in Equation 1, the change in the differentialsense output voltage is: $\begin{matrix}\begin{matrix}{{\frac{{\Delta \quad V_{B}} - {\Delta \quad V_{A}}}{V_{m\quad p} - V_{m\quad n}} \cdot \frac{2\quad C_{0}}{\delta}} = {\frac{{S_{I\quad t}\lambda_{l\quad t}} + {S_{r\quad b}\lambda_{r\quad b}}}{\lambda_{lt} + \lambda_{rb} + \frac{C_{p_{A}}}{2\quad C_{0}}} - \frac{{S_{I\quad b}\lambda_{l\quad b}} + {S_{r\quad t}\lambda_{r\quad t}}}{\lambda_{l\quad b} + \lambda_{rt} + \frac{C_{p_{B}}}{2\quad C_{0}}}}} \\{= \frac{\begin{matrix}\begin{matrix}{{\lambda_{lt}{\lambda_{l\quad b}\left( {S_{I\quad t} - S_{Ib}} \right)}} + {\lambda_{rb}{\lambda_{l\quad b}\left( {S_{rb} - S_{l\quad b}} \right)}} +} \\{{\lambda_{lt}{\lambda_{r\quad t}\left( {S_{l\quad t} - S_{r\quad t}} \right)}} + {\lambda_{rb}{\lambda_{r\quad t}\left( {S_{rb} - S_{r\quad t}} \right)}} +}\end{matrix} \\{{\frac{C_{p_{B}}}{2\quad C_{0}}\left( {{S_{I\quad t}\lambda_{l\quad t}} + {S_{r\quad b}\lambda_{r\quad b}}} \right)} - {\frac{C_{p_{A}}}{2\quad C_{0}}\left( {{S_{I\quad b}\lambda_{l\quad b}} + {S_{r\quad t}\lambda_{r\quad t}}} \right)}}\end{matrix}}{\left( {\lambda_{lt} + \lambda_{rb} + \frac{C_{p_{A}}}{2\quad C_{0}}} \right)\left( {\lambda_{l\quad b} + \lambda_{r\quad t} + \frac{C_{p_{B}}}{2\quad C_{0}}} \right)}}\end{matrix} & (2)\end{matrix}$

[0053] Using the expressions in Equation 2, we represent the states ofthe capacitive network (shown in FIG. 7) in the form of a truth table inTable 3. In Table 3, the most interesting modes are those that have aneven number of 1's. Four such modes exist, all of which have been namedas shown. TABLE 2 Comb capacitance values at nominal (assuming noDC-offset) and displaced positions. Capacitance Value at nominalposition Value at displaced position C_(1lt) C₀λ_(lt) (C₀ − δ)λ_(lt)C_(2lt) C₀λ_(lt) (C₀ + δ)λ_(lt) C_(1rb) C₀λ_(rb) (C₀ − δ)λ_(rb) C_(2rb)C₀λ_(rb) (C₀ + δ)λ_(rb) C_(1lb) C₀λ_(lb) (C₀ − δ)λ_(lb) C_(2lb) C₀λ_(lb)(C₀ + δ)λ_(lb) C_(1rt) C₀λ_(rt) (C₀ − δ)λ_(rt) C_(2rt) C₀λ_(rt) (C₀ +δ)λ_(rt)

[0054] TABLE 3 Modulation states of comb capacitances. Dipole statesNormalized sense outputs C_(1lg):C_(2lt) C_(1rb):C_(2rb) C_(1lb):C_(2lb)C_(1rt):C_(2rt) (ΔV_(B) − ΔV_(A))/(V_(mp) − V_(mn)) Mode names 0 0 0 0${\delta \frac{\lambda_{lt} + \lambda_{rb}}{{2{C_{0}\left( {\lambda_{lt} + \lambda_{rb}} \right)}} + C_{P_{A}}}} - {\delta \frac{\lambda_{lb} + \lambda_{rt}}{{2{C_{0}\left( {\lambda_{lb} + \lambda_{rt}} \right)}} + C_{P_{B}}}}$

Self-test XY-mode 0 0 0 1${\delta \frac{\lambda_{lt} + \lambda_{rb}}{{2{C_{0}\left( {\lambda_{lt} + \lambda_{rb}} \right)}} + C_{P_{A}}}} - {\delta \frac{\lambda_{lb} - \lambda_{rt}}{{2{C_{0}\left( {\lambda_{lb} + \lambda_{rt}} \right)}} + C_{P_{B}}}}$

0 0 1 1${\delta \frac{\lambda_{lt} + \lambda_{rb}}{{2{C_{0}\left( {\lambda_{lt} + \lambda_{rb}} \right)}} + C_{P_{A}}}} + {\delta \frac{\lambda_{lb} - \lambda_{rt}}{{2{C_{0}\left( {\lambda_{lb} + \lambda_{rt}} \right)}} + C_{P_{B}}}}$

0 0 1 1${\delta \frac{\lambda_{lt} + \lambda_{rb}}{{2{C_{0}\left( {\lambda_{lt} + \lambda_{rb}} \right)}} + C_{P_{A}}}} + {\delta \frac{\lambda_{lb} + \lambda_{rt}}{{2{C_{0}\left( {\lambda_{lb} + \lambda_{rt}} \right)}} + C_{P_{B}}}}$

Normal sense mode 0 1 0 0${\delta \frac{\lambda_{lt} - \lambda_{rb}}{{2{C_{0}\left( {\lambda_{lt} + \lambda_{rb}} \right)}} + C_{P_{A}}}} - {\delta \frac{\lambda_{lb} + \lambda_{rt}}{{2{C_{0}\left( {\lambda_{lb} + \lambda_{rt}} \right)}} + C_{P_{B}}}}$

0 1 0 1${\delta \frac{\lambda_{lt} - \lambda_{rb}}{{2{C_{0}\left( {\lambda_{lt} + \lambda_{rb}} \right)}} + C_{P_{A}}}} - {\delta \frac{\lambda_{lb} - \lambda_{rt}}{{2{C_{0}\left( {\lambda_{lb} + \lambda_{rt}} \right)}} + C_{P_{B}}}}$

Self-test Y-mode 0 1 1 0${\delta \frac{\lambda_{lt} - \lambda_{rb}}{{2{C_{0}\left( {\lambda_{lt} + \lambda_{rb}} \right)}} + C_{P_{A}}}} + {\delta \frac{\lambda_{lb} - \lambda_{rt}}{{2{C_{0}\left( {\lambda_{lb} + \lambda_{rt}} \right)}} + C_{P_{B}}}}$

Self-test X-mode 0 1 1 1${\delta \frac{\lambda_{lt} - \lambda_{rb}}{{2{C_{0}\left( {\lambda_{lt} + \lambda_{rb}} \right)}} + C_{P_{A}}}} + {\delta \frac{\lambda_{lb} + \lambda_{rt}}{{2{C_{0}\left( {\lambda_{lb} + \lambda_{rt}} \right)}} + C_{P_{B}}}}$

[0055] The combined effect of all local manufacturing variations on adependent variable (like capacitance C distributed over an area) can berepresented by $\begin{matrix}{C = {C_{00}{\int_{A}^{\quad}{{\lambda \left( {\alpha,\beta} \right)}\quad \frac{{\alpha}{\beta}}{A}}}}} & (3)\end{matrix}$

[0056] where α and β are length parameters in the area A. Note that intotal absence of local variations,

λ(α,β)=1,∀α,∀β,

[0057] which reduces Eq. 3 to the simple nominal relation,

C=C₀₀

[0058] We assume a variable separable relation,

λ(α,β)=[1+X(α)][1+Y(β)]  (4)

[0059] which implies that local variations along X and Y axes areindependent of each other.

[0060] The condition

X(α)=Y(β)=0,∀α,∀β

[0061] implies no local variation. Note that α=β=0 at the center of thelayout. Using Eq. 4 in Eq. 3, the following expression for capacitanceis obtained: $\begin{matrix}\begin{matrix}{C = {C_{00}{\int_{A}^{\quad}{{\left\lbrack {1 + {X(\alpha)}} \right\rbrack \left\lbrack {1 + {Y(\beta)}} \right\rbrack}\frac{{\alpha}{\beta}}{A}}}}} \\{{= {C_{00}{\int_{\alpha_{1}}^{\alpha_{2}}{\left\lbrack {1 + {X(\alpha)}} \right\rbrack \quad \frac{\alpha}{\alpha_{2} - \alpha_{1}}{\int_{\beta_{1}}^{\beta_{2}}{\left\lbrack {1 + {Y(\beta)}} \right\rbrack \quad \frac{\beta}{\beta_{2} - \beta_{1}}}}}}}},} \\{{{{where}\quad A} = {\left( {\alpha_{2} - \alpha_{1}} \right)\left( {\beta_{2} - \beta_{1}} \right)}}} \\{= {{C_{00}\left\lbrack {1 + \overset{\_}{X}} \right\rbrack}\left\lbrack {1 + \overset{\_}{Y}} \right\rbrack}}\end{matrix} & (5)\end{matrix}$

[0062] We will use the above expression for representing combcapacitances in our analysis. The above expression uses the concept ofaverage manufacturing variations because all signals are generated fromthe interaction of distributed capacitances.

[0063] Also, we assume

C_(PA=C) _(PB=C) _(P)

[0064] (i.e., same parasitic capacitance at each of the differentialsense output nodes, A and B). From these assumptions and Equation 2, thedifferential sense outputs for each mode are computed and tabulated inTable 4. TABLE 4 All modes λ_(lb)(α⁻, β⁻) = [1 + X(α⁻)]  [1 + Y(β⁻)]

λ_(lt)(α⁻, β⁺) = [1 + X(α⁻)]  [1 + Y(β⁺)]

λ_(rt)(α⁺, β⁺) = [1 + X(α⁺)]  [1 + Y(β⁺)]

λ_(lb)(α⁺, β⁻) = [1 + X(α⁺)]  [1 + Y(β⁻)]

$\mu = \frac{C_{p}}{4C_{0}}$

X_(s) = X(α⁺) + X(α⁻), Y_(s) = Y(β⁺) + Y(β⁻)

X_(d) = X(α⁺) − X(α⁻), Y_(d) = Y(β⁺) − Y(β⁻)

${H\left( {X_{s},Y_{s},X_{d},Y_{d}} \right)} = {\left\lbrack {{2\mu} + \frac{\left( {2 + X_{s}} \right)\left( {2 + Y_{s}} \right)}{2}} \right\rbrack^{2} - \left( \frac{X_{d}Y_{d}}{2} \right)^{2}}$

Mode 0011${{\Delta V}_{B} - {\Delta V}_{A}} = {\left( {V_{mp} - V_{mn}} \right){\frac{\delta}{C_{0}}\left\lbrack {1 - {\mu \cdot \frac{{\left( {2 + X_{s}} \right)\left( {2 + Y_{s}} \right)} + {4\mu}}{H\left( {X_{s},Y_{s},X_{d},Y_{d}} \right)}}} \right\rbrack}}$

Mode 0110${{\Delta V}_{B} - {\Delta V}_{A}} = {\left( {V_{mp} - V_{mn}} \right)\frac{\delta}{2C_{0}}{\left( {- X_{d}} \right)\left\lbrack \frac{{2\left\{ {1 + Y_{s} + {{Y\left( \beta^{+} \right)}{Y\left( \beta^{-} \right)}}} \right\} \left( {2 + X_{s}} \right)} + {2{\mu Y}_{s}} + {4\mu}}{H\left( {X_{s},Y_{s},X_{d},Y_{d}} \right)} \right\rbrack}}$

Mode 0101${{\Delta V}_{B} - {\Delta V}_{A}} = {\left( {V_{mp} - V_{mn}} \right)\frac{\delta}{2C_{0}}{\left( {+ X_{d}} \right)\left\lbrack \frac{{2\left\{ {1 + X_{s} + {{X\left( \alpha^{+} \right)}{X\left( \alpha^{-} \right)}}} \right\} \left( {2 + Y_{s}} \right)} + {2X_{s}} + {4\mu}}{H\left( {X_{s},Y_{s},X_{d},Y_{d}} \right)} \right\rbrack}}$

Mode 0000${{\Delta V}_{B} - {\Delta V}_{A}} = {\left( {V_{mp} - V_{mn}} \right)\frac{\delta}{2C_{0}}{\left( {{- X_{d}}Y_{d}} \right)\left\lbrack \frac{2\mu}{H\left( {X_{s},Y_{s},X_{d},Y_{d}} \right)} \right\rbrack}}$

[0065] IF X(α) and Y(β) are odd functions, we have

X_(s)=Y_(s)=0.

[0066] Also, for a normal manufacturing process,$\left( \frac{X_{d}Y_{d}}{4} \right)^{2}{\operatorname{<<}1}$

[0067] Under such conditions, the expressions of sensitivities of themodes in Table 4 reduce to the ones listed in Table 5. TABLE 5 Mode(ΔV_(B) − ΔV_(A))/(V_(mp) − V_(mn)) Dependence 0011$\frac{\delta}{C_{0}}\left\lbrack \frac{1}{1 + \frac{C_{p}}{4C_{0}}} \right\rbrack$

Independent of X and Y variations 0110$\frac{\delta}{2C_{0}}{\left( {- X_{d}} \right)\left\lbrack \frac{1 + \frac{C_{p}}{4C_{0}} + {{Y\left( \beta^{+} \right)}{Y\left( \beta^{-} \right)}}}{\left( {1 + \frac{C_{p}}{4C_{0}}} \right)^{2}} \right\rbrack}$

Dependent on X variations only 0101$\frac{\delta}{2C_{0}}{\left( {+ Y_{d}} \right)\left\lbrack \frac{1 + \frac{C_{p}}{4C_{0}} + {{X\left( \alpha^{+} \right)}{X\left( \alpha^{-} \right)}}}{\left( {1 + \frac{C_{p}}{4C_{0}}} \right)^{2}} \right\rbrack}$

Dependent on Y variations only 0000$\frac{\delta}{4C_{0}}{\left( {{- X_{d}}Y_{d}} \right)\left\lbrack \frac{\frac{C_{p}}{4C_{0}}}{\left( {1 + \frac{C_{p}}{4C_{0}}} \right)^{2}} \right\rbrack}$

Dependent on both X and Y variations

[0068] Simulation Results TABLE 6 Parameter Value Unit Resonantfrequency (f_(x)) 12.5 kHz Sensor sensitivity 0.88 mV/G Modulationvoltage amplitude 5.0 V Actuation voltage amplitude 1.5 V Input referrednoise 100 μG/{square root}Hz Bandwidth of baseband sense signal 500 Hz

[0069] A CMOS-MEMS accelerometer with the parameters listed in Table 6is modified to include the necessary characteristics to implement thedisclosed BIST approach. Specifically, switches are used to interchangethe polarity of modulation signals applied to finger pairs [S₃S₇] and[S₄S₈], to control the application of modulation signals to the dummyfingers [D₁ D₄], and to select one of the two self-test differencesignals if only one output pin is reserved for self-test. Simulationexperiments are performed to examine the capability of this approach todetect asymmetry caused by: (i) a single dielectric particle acting as abridge between a pair of structures where at least one is movable; (ii)a variation in vertical misalignment between fixed and movable fingerscaused by curl mismatch; (iii) a variation in local etch; and (iv)unequal parasitics in the interconnects from the self-test sense pointsto the differential sense amplifier.

[0070] With respect to asymmetry caused by a single particle acting as abridge, particles can originate from the clean room but also from theremoval of the sacrificial layer during the release step. Particlesformed out of the sacrificial layer can be as large as a few μm and aretherefore large enough to act as bridges between structures. Simulationexperiments were conducted using NODAS [See Jing et al., “CMOSMicromechanical Bandpass Filter Design Using a Hierarchical MEMS CircuitLibrary,” Proc. of Micro Electro Mechanical Systems Conference, pp.187-192 (January 2000)], and AHDL (Analog Hardware Descriptive Language)simulator for mixed-domain circuits. NODAS has been shown to closelymatch experimental result. The efficacy of NODAS as a reliable and muchfaster simulator than finite element analysis has also beendemonstrated. Both the electromechanical microstructure and electroniccircuitry are simulated together. The electronic circuitry is based on adesign [See Wu et al., “A Low-Noise Low-Offset Chopper-StabilizedCapacitive-readout Amplifier for CMOS MEMS Accelerometers,” Proc. ofInternational Solid State Circuits Conference, pp. 428-429 (February2002)] that has been fabricated and validated with CMOS-MEMS devices.

[0071] One of the parameters used to decide pass/fail for anaccelerometer is its resonant frequency (f_(x)) for translation in the Xdirection. The acceptable range for resonant frequency includes amaximum deviation of 25% from the nominal value, which translates to arange of 94 kHz to 156 kHz for the accelerometer design of Table 7. Theacceptable range for the normal sense signal is 20% from the nominalvalue (105 mV), which implies a range of 84 mV to 126 mV. If thebandwidth of the processed sense signal is restricted to about 500 Hzthen a reasonably low noise voltage floor of about 2 μV can be achieved.TABLE 7 Self-test finger outputs Self-test beam outputs Defect location(%) f_(x)(kHz) Normal sense output (mv) $\begin{matrix}V_{l}^{f} \\({mV})\end{matrix}\quad$

$\begin{matrix}V_{r}^{f} \\({mV})\end{matrix}\quad$

$\begin{matrix}{V_{l}^{f} - V_{r}^{f}} \\({\mu V})\end{matrix}\quad$

${\begin{matrix}{1 - \frac{V_{l}^{f}}{V_{r}^{f}}} \\(\%)\end{matrix}\quad}\quad$

$\begin{matrix}V_{l}^{b} \\({mV})\end{matrix}\quad$

$\begin{matrix}V_{r}^{b} \\({mV})\end{matrix}\quad$

$\begin{matrix}{V_{l}^{b} - V_{r}^{b}} \\({\mu V})\end{matrix}\quad$

${\begin{matrix}{1 - \frac{V_{l}^{b}}{V_{r}^{b}}} \\(\%)\end{matrix}\quad}\quad$

None 12.53 10.52 5.256 5.259 −3 0 1.851 1.853 −2 0 10 13.12 9.58 4.7854.788 −3 0 1.643 1.670 −27 1.6 20 14.00 8.40 4.195 4.197 −2 0 1.3381.411 −73 5.2 30 15.32 6.98 3.490 3.492 −2 0 1.016 1.149 −133 11.5

[0072] TABLE 8 Self-test finger outputs Self-test beam outputs Defectlocation (%) f_(x)(kHz) Normal sense output (mv) $\begin{matrix}V_{l}^{f} \\({mV})\end{matrix}\quad$

$\begin{matrix}V_{r}^{f} \\({mV})\end{matrix}\quad$

$\begin{matrix}{V_{l}^{f} - V_{r}^{f}} \\({\mu V})\end{matrix}\quad$

${\begin{matrix}{1 - \frac{V_{l}^{f}}{V_{r}^{f}}} \\(\%)\end{matrix}\quad}\quad$

$\begin{matrix}V_{l}^{b} \\({mV})\end{matrix}\quad$

$\begin{matrix}V_{r}^{b} \\({mV})\end{matrix}\quad$

$\begin{matrix}{V_{l}^{b} - V_{r}^{b}} \\({\mu V})\end{matrix}\quad$

${\begin{matrix}{1 - \frac{V_{l}^{b}}{V_{r}^{b}}} \\(\%)\end{matrix}\quad}\quad$

None 12.53 10.52 5.256 5.259 −3 0 1.851 1.853 −2 0  0 13.74 8.30 4.5443.761 783 20.8 1.539 1.541 −2 0 10 14.09 7.85 4.347 3.499 848 24.2 1.4651.4661 −1 0 20 14.57 7.27 4.089 3.183 906 28.5 1.370 1.371 −1 0

[0073] TABLE 9 Self-test finger outputs Self-test beam outputsδH_(right)(μm) f_(x)(kHz) Normal sense output (mv) $\begin{matrix}V_{l}^{f} \\({mV})\end{matrix}\quad$

$\begin{matrix}V_{r}^{f} \\({mV})\end{matrix}\quad$

$\begin{matrix}{V_{l}^{f} - V_{r}^{f}} \\({\mu V})\end{matrix}\quad$

${\begin{matrix}{1 - \frac{V_{l}^{f}}{V_{r}^{f}}} \\(\%)\end{matrix}\quad}\quad$

$\begin{matrix}V_{l}^{b} \\({mV})\end{matrix}\quad$

$\begin{matrix}V_{r}^{b} \\({mV})\end{matrix}\quad$

$\begin{matrix}{V_{l}^{b} - V_{r}^{b}} \\({\mu V})\end{matrix}\quad$

${\begin{matrix}{1 - \frac{V_{l}^{b}}{V_{r}^{b}}} \\(\%)\end{matrix}\quad}\quad$

0 12.53 10.52 5.256 5.259 −3 0 1.851 1.853 −2 0 +0.5 12.51 10.44 5.2485.193 55 −1.0 1.841 1.836 5 −0.3 +1.0 12.51 10.23 5.224 5.001 223 −4.41.810 1.789 21 −1.2 +1.5 12.51 9.87 5.177 4.695 482 −10.2 1.760 1.712 48−2.8 +2.0 12.51 9.42 5.115 4.309 806 −18.7 1.696 1.616 80 −4.9

[0074] In the disclosed BIST approach to the accelerometer, self-testoutputs are produced from normal sense fingers and spring beams.Depending on the particular nature of an asymmetry, one output may bemore suited than the other at observing the effects of a defect. Also,the asymmetries detected at one output need not be a subset of thosedetected at the other. Hence, the use of self-test outputs from bothbeams and fingers, and possibly other sites, is necessary to minimizedefective parts from being shipped as good parts.

[0075] In the following subsections, the symbols V_(bl) and V_(br), areused to refer to the self-test outputs sampled from the spring beams B₁and B₂ (see FIG. 2), respectively. Similarly, V_(fl) and V fr representself-test outputs sampled from the regular sense finger pairs [M₅M₇] and[M₆M₈] (see FIG. 2), respectively. The voltage difference between eachpair of self-test signals must be more than the 2 μV noise floor to beconsidered significant.

[0076] Beam Bridges

[0077] A bridge defect can be caused by particulate matter that attachesa movable beam to an adjacent structure (e.g, a dummy finger) therebyhindering its motion. Due to the four-fold symmetry of theaccelerometer, simulation of a bridge defect has been limited to onequadrant of the layout. Specifically, beam B₁ of the upper left quadrantof FIG. 2 is used. Column 1 of Table 7 indicates the defect locationexpressed as a percentage of beam length. The 0% point is the anchoredend of the beam and the 100% point is where the beam meets the shuttle.Columns 2 and 3 list the values of resonant frequency and normal senseoutput, respectively, for bridge defects located at different locationsalong the beam.

[0078] As the defect location moves from the anchor end of the beam tothe end where it is attached to the shuttle, the beam stiffnessincreases. Consequently, shuttle displacement decreases. Also the layoutasymmetry becomes more pronounced resulting in an increased differencebetween the two beam sense outputs V_(bl) and V_(br).

[0079] The listed resonant frequency values shown in Table 7 are allwithin the acceptable range, indicating these defects will pass aresonant frequency test. The normal sense output is outside itsacceptable range only for the 30% point. Hence, in a majority of cases,a test based on resonant frequency and normal sense output will beineffective. The finger sense outputs V_(fl) and V_(fr) hardly divergebecause of the stiffness of the shuttle which means virtually equaldisplacements on both sides. However, the beam self-test outputs V_(bl)and V_(br) do indicate the presence of an asymmetry.

[0080] Finger Bridges

[0081] A finger bridge defect is similar to a beam bridge defect exceptthat it is located between a movable finger and a fixed finger.Naturally, it acts as a hindrance to shuttle motion. In this analysisthe material of the bridging defect has been assumed to be dielectricbecause such defects are harder to detect. As before, the symmetry ofthe accelerometer is used to limit simulations to the upper rightquadrant of the layout. An inter-finger defect that bridges fingers M₆and S₃ in the upper right quadrant of FIG. 2 is considered. Aninter-finger bridge defect is modeled using the approach in N. Deb andR. D. Blanton, “High-Level Fault Modeling in Surface-MicromachinedMEMS,” Proc. of Design, Test, Integration, and Packaging of MEMS/MOEMS,pp. 228-235 (May 2000). Defect location is expressed as a percentage ofmovable finger length. The 0% point is the movable finger tip, and the100% point is the movable finger base where it meets the shuttle.

[0082] The results in Table 8 indicate that a finger bridge defect maypass a resonant frequency test but will fail a sensitivity test.However, the normal sense output by itself does not indicate anasymmetry. The beam self-test outputs V_(bl) and V_(br) hardly divergeand hence are ineffective. However, the finger self-test outputs V_(fl)and V_(fr) diverge significantly and clearly indicate the presence ofasymmetry.

[0083] Finger Height Mismatch

[0084] Ideally, the fingers should all be at the same height above thedie surface. But variations in parameters such as temperature andresidual stress lead to finger height mismatch. Height mismatch betweenthe fixed and movable fingers reduces inter-finger overlap and henceinter-finger capacitance. Here it is shown how left-right asymmetrycaused by such height mismatch can be detected by this BIST approach.Without loss of generality, the finger height mismatch is assumed toexist on the right side of the accelerometer only (δH_(left).0). Table 9lists the simulation results. Column 1 lists the relative heightmismatch which is expressed as δH_(right).δH_(left).δH_(right). Resultsfor negative values of mismatch have not been separately simulated sincethey would yield similar results. Such symmetrical behavior is exhibitedby CMOS-MEMS because the gap between the substrate and the accelerometerfingers is 20 μm.

[0085] With increasing height mismatch, the difference in sensingbetween the two sides increases, as evident from both [V_(bl)V_(br)] and[V_(fl) V_(fr)]. The resonant frequency remains virtually unchangedclearly indicating a resonant frequency test will not detect this formof asymmetry. The normal sense output reveals an acceptable reducedvoltage due to the reduced sense capacitance. Hence, a test based onresonant frequency and normal sense output will be ineffective indetecting the asymmetry. However, the difference between the fingerself-test outputs V_(fl) and V_(fr) clearly indicate the presence ofasymmetry. Although not as sensitive, the beam self-test outputs V_(bl)and V_(br) also vary with the amount of mismatch and therefore indicatean asymmetry as well.

[0086] Local Etch Variations TABLE 10 Etch Self-test finger outputsSelf-test beam outputs variation (μm) f_(x)(kHz) Normal sense output(mv) $\begin{matrix}V_{l}^{f} \\({mV})\end{matrix}\quad$

$\begin{matrix}V_{r}^{f} \\({mV})\end{matrix}\quad$

$\begin{matrix}{V_{l}^{f} - V_{r}^{f}} \\({\mu V})\end{matrix}\quad$

${\begin{matrix}{1 - \frac{V_{l}^{f}}{V_{r}^{f}}} \\(\%)\end{matrix}\quad}\quad$

$\begin{matrix}V_{l}^{b} \\({mV})\end{matrix}\quad$

$\begin{matrix}V_{r}^{b} \\({mV})\end{matrix}\quad$

$\begin{matrix}{V_{l}^{b} - V_{r}^{b}} \\({\mu V})\end{matrix}\quad$

${\begin{matrix}{1 - \frac{V_{l}^{b}}{V_{r}^{b}}} \\(\%)\end{matrix}\quad}\quad$

+0.025 12.33 10.30 5.404 4.897 507 −10.3 1.837 1.789 48 −2.7 +0.02012.37 10.34 5.375 4.967 408 −8.2 1.840 1.801 39 −2.2 +0.010 12.45 10.435.315 5.110 205 −4.0 1.845 1.826 19 −1.0 0 12.53 10.52 5.256 5.259 −3 01.851 1.853 −2 0 −0.010 12.61 10.61 5.197 5.414 −217 4.0 1.858 1.881 −231.2 −0.020 12.69 10.71 5.136 5.574 −438 7.9 1.865 1.910 −45 2.3 −0.02512.73 10.76 5.105 5.656 −551 9.7 1.869 1.925 −56 2.9

[0087] TABLE 11 Parasitic Self-test finger outputs Self-test beamoutputs mismatch (%) f_(x)(kHz) Normal sense output (mv) $\begin{matrix}V_{l}^{f} \\({mV})\end{matrix}\quad$

$\begin{matrix}V_{r}^{f} \\({mV})\end{matrix}\quad$

$\begin{matrix}{V_{l}^{f} - V_{r}^{f}} \\({\mu V})\end{matrix}\quad$

${\begin{matrix}{1 - \frac{V_{l}^{f}}{V_{r}^{f}}} \\(\%)\end{matrix}\quad}\quad$

$\begin{matrix}V_{l}^{b} \\({mV})\end{matrix}\quad$

$\begin{matrix}V_{r}^{b} \\({mV})\end{matrix}\quad$

$\begin{matrix}{V_{l}^{b} - V_{r}^{b}} \\({\mu V})\end{matrix}\quad$

${\begin{matrix}{1 - \frac{V_{l}^{b}}{V_{r}^{b}}} \\(\%)\end{matrix}\quad}\quad$

0 12.53 10.52 5.256 5.259 −3 −0 1.851 1.853 −2 0 1.25 12.51 10.50 5.2395.265 −26 0.5 1.843 1.850 −7 0.4 2.5 12.51 10.49 5.221 5.270 −49 0.91.836 1.847 −11 0.6 3.75 12.51 10.48 5.203 5.274 −71 1.3 1.828 1.845 −170.9 5.0 12.51 10.47 5.186 5.280 −94 1.8 1.819 1.841 −22 1.2

[0088] An etching process is used in fabrication to remove sacrificialmaterial to free the micromechanical sensor. Material removal through anetching process varies with time and space even though such variation isnot desirable. For example, a rectangular structure designed to havelength l and width w may be subjected to more than the intended etch bya length δ, resulting in dimensions [l-2δ,w-2δ]. This is due to the factthat each side-wall of the rectangular structure shifts inwards by δ sothat each dimension reduces by 2δ. This type of etch variation is calledover-etch. In a similar fashion, under-etch causes an oversize structureof size [l+2δ, w+2δ].

[0089] Etch variation can also be local in nature. Consider tworectangular structures that are designed to be identical but duringfabrication they are subjected to different etch variations, δ₁ and δ₂.As a result, the two structures will have different dimensions, causinga left-right asymmetry.

[0090] Without loss of generality, we assumed in simulation that theaccelerometer's left side has nominal etch while the right side haseither over- or under-etch. Table 10 gives the simulation results.Column 1 lists the relative etch mismatch. The mismatch in etchvariation is positive when the right side is more etched than the leftside. As the relative over-etch increases, the sensitivity of the rightside reduces because of the loss in the inter-finger capacitance.Consequently, the differences |V_(fl)−V_(fr)| and |V_(bl)−V_(br)|increase. For increasing levels of relative under-etch, twocounteracting effects become significant. The increased beam thicknesson the right side causes increased stiffness which in turn reducesdisplacement. However, the reduced inter-finger gap causes higherinter-finger capacitance which more than offsets the loss ofsensitivity. In any case, a higher level of local etch variation leadsto a greater difference between each pair of self-test outputs.

[0091] A test based on resonant frequency test and normal sense outputwill not detect the presence of this type of asymmetry because bothparameters are within their respective acceptable ranges. The fingerself-test outputs V_(fl) and V_(fr) diverge and hence indicate anasymmetry. The beam self-test outputs V_(bl) and V_(br) indicate anasymmetry as well. As in the case of finger height mismatch,|V_(fl)−V_(fr)|>10|V_(bl)−V_(br)|, which means the finger self-testoutputs are stronger indicators of this type of asymmetry as compared tothe beam self-test outputs.

[0092] Parasitic Variation

[0093] Ideally, the sensing circuitry for self-test should only besensitive to asymmetries in the micromechanical sensor and not theexternal electronics (including interconnects). In reality, a differencesignal may be due to variation external to the sensor area, such as inthe interconnects which carry the self-test sense signals to the inputsof the differential amplifier. With reference to the differentialamplifier in FIG. 4, assume the interconnect capacitances (C_(pl) andC_(p2)) are unequal. The objective is to determine the extent to whichthe parasitic capacitance mismatch due to such interconnect asymmetrywill produce a significant sense amplifier output. The maximum value ofthe parasitic mismatch can be used to decide a suitable threshold fordetection of sensor asymmetry during self-test. The nominal interconnectcapacitance is assumed to be 40 fF. A maximum mismatch of 5% between thetwo interconnects is considered, assuming that a good layout design anda stable process can restrict such variations to the presumed limit.Without loss of generality, we assumed that the interconnect from theleft sense points (for both the finger and beam outputs) are larger thanthe right. The simulation results are listed in Table 11.

[0094] The parasitic mismatch has a more pronounced effect for the beamself-test outputs [V_(bl)V_(br)] as compared to the finger self-testoutputs [V_(fl) V_(fr)] because the beam self-test signals are muchweaker. It is observed that V_(fl.) V_(fr.) is less sensitive tovariations in the interconnect when compared to variation in local etchand finger height. However, the same is not true for the beam self-testoutputs. Interconnect capacitance mismatch does indeed cause self-testoutputs to exceed the noise floor. However, in a majority of the casesconsidered, the output magnitude does not rival that produced by theother defects. This implies that variations of up to 5% in theinterconnect capacitance are unlikely to falsely indicate asymmetry inthe micromechanical structure of the accelerometer. Only a beam bridgedefect that is located close to its anchor point will produce a beamself-test output difference that is not significantly larger than thatproduced by interconnect mismatch.

[0095] BIST Implementation

[0096] Because simulation alone is not sufficient for validating thedifferential BIST approach disclosed here, a prototype was designed andfabricated in which a subset of the BIST features was added to theaccelerometer design of Wu et al., “A Low-Noise Low-OffsetChopper-Stabilized Capacitive-readout Amplifier for CMOS MEMSAccelerometers,” supra. The BIST implementation was limited to fingerself-test for the purpose of validation. The layout of the prototypedesign is shown in FIG. 8. The technology used is athree-metal-one-poly, 06 μm CMOS process.

[0097] The control circuitry for BIST is illustrated in FIG. 9. The twophases of the modulation voltage signal are V_(mp) and V_(mn). Thedigital self-test input pin (SI) is driven to a logic zero during normaloperation. For SI=0, the capacitive network reduces to the one shown inFIG. 5A. Self-test is activated when SI=1. For this case, the capacitivenetwork reduces to the one shown in FIG. 5B. The overhead of the BISTcircuit includes routing area for two extra wires for the modulationsignals and area for simple 2-by-2 cross-bar switch. The area of theswitch is 105 μm by 35 μm, which is 0.06% of the of the total die area(25 mm by 25 mm). The routing for the two extra modulation lines in thesensor was achieved by using the POLY1 layer. An extra input pin isrequired for activating the self-test. However, no extra pin is requiredfor the self-test output because the same pin is used for normal andBIST operation.

[0098] The die size available to us allowed us to implement threeidentical micromechanical sensors. The same self-test control circuitry,consisting of two switches (SW1 and SW2 in FIG. 9), was used to controlall three sensors simultaneously. In other words, the self-test mode isactivated for all the sensors in parallel. Because the same output pinis shared by the sense signals from the three sensors, anotherswitch-based control block is used to ensure that only one sense signalis transferred to the output pin at any given time. The control blockuses high-impedance circuitry to isolate the remaining two sensor outputsignals.

[0099] The differential self-test method described in this disclosure isfocused on enhancing observation and therefore complements the existingbuilt-in stimulus generation techniques found in industry and proposedin the literature. We have demonstrated the ability to detect thepresence of various defect types that cause local asymmetry, which arenot detectable by typical specification based tests that measure bothresonant frequency and sensitivity. The self-test method describedherein can substantially enhance detection of hard-to-detect defects.The tradeoff of such an approach is a modest increase in designcomplexity and device area for additional electronics.

[0100] While the present disclosure has been described in connectionwith various embodiments, those of ordinary skill in the art willrecognize that various modifications and variations are possible. Thisdisclosure and the following claims are intended to cover all suchmodifications and variations.

What is claimed is:
 1. In a MEMS device, the improvement comprising: aplurality of sensors electrically isolated from one another andpositioned to produce signals of substantially identicalcharacteristics; and circuitry responsive to said plurality of sensorsfor comparing said signals produced by said plurality of sensors.
 2. TheMEMS device of claim 1 additionally comprising circuitry for actuatingthe MEMS device.
 3. The MEMS device of claim 1 wherein said MEMS deviceis implemented using batch-fabrication techniques, and wherein saidcircuitry and connections between said circuitry and said sensors areimplemented using batch-fabrication techniques.
 4. The MEMS device ofclaim 1 wherein said MEMS device is selected from the group consistingof resonators, accelerometers, gyroscopes, antennas, micromotors and inkjet print head Microsystems.
 5. A MEMS device, comprising: a pluralityof fixed beams arranged symmetrically; a plurality of movable beamsarranged symmetrically; a first sensor formed by certain of said fixedand movable beams; a second sensor, electrically isolated from saidfirst sensor, and formed by at least certain other of said fixed andmovable beams; and a circuit responsive to said first and second sensorsfor comparing signals produced by said first and second sensors.
 6. Thedevice of claim 5 additionally comprising circuitry for actuating saidplurality of movable beams.
 7. The device of claim 5 wherein said MEMSdevice is implemented using batch-fabrication techniques, said circuitand connections between said circuit and said sensors are implementedusing batch-fabrication techniques.
 8. In a symmetric MEMS device, theimprovement comprising: a plurality of sensors positioned to producesignals of substantially identical characteristics; and circuitryresponsive to said plurality of sensors for real time comparison of saidsignals produced by said plurality of sensors.
 9. The MEMS device ofclaim 8 additionally comprising circuitry for actuating the MEMS device.10. The MEMS device of claim 8 wherein said MEMS device is implementedusing batch-fabrication techniques, said circuitry and connectionsbetween said circuitry and said sensors are implemented usingbatch-fabrication techniques.
 11. The MEMS device of claim 8 whereinsaid MEMS device is selected from the group consisting of resonators,accelerometers, gyroscopes, antennas, micromotors and ink jet print headMicrosystems.
 12. A MEMS device, comprising: a plurality of fixed beamsarranged symmetrically; a plurality of movable beams arrangedsymmetrically; a first sensor formed by certain of said fixed andmovable beams; a second sensor formed by at least certain other of saidfixed and movable beams; and a circuit responsive to said first andsecond sensors for real time comparison of said signals produced by saidsensors.
 13. The MEMS device of claim 12 additionally comprisingcircuitry for actuating said plurality of movable beams.
 14. The MEMSdevice of claim 12 wherein said MEMS device is implemented usingbatch-fabrication techniques, said circuit and connections between saidcircuit and said sensors are implemented using batch-fabricationtechniques.
 15. A method, comprising: actuating a MEMS device; andcomparing the outputs from a first and a second sensor electricallyisolated from one another and positioned to produce signals ofsubstantially identical characteristics.
 16. The method of claim 15wherein said actuating is performed mechanically.
 17. The method ofclaim 15 wherein said actuating is performed electrically.
 18. Themethod of claim 17 wherein said electrically actuating comprisesinterchanging the polarity of a modulation signal applied between pairsof fixed and movable members.
 19. The method of claim 15 wherein saidcomparing is performed in real time.
 20. The method of claim 15 whereinsaid comparing reveals local asymmetry.
 21. The method of claim 20wherein said local asymmetry includes one of a particle bridge, verticalmisalignment, variation in local etch and unequal parasitics in theinterconnects between the sensors and the circuit for analyzing.
 22. Amethod, comprising: actuating a MEMS device; and comparing the outputsfrom a first and a second symmetrically located sensor in real time. 23.The method of claim 22 wherein said actuating is performed mechanically.24. The method of claim 22 wherein said actuating is performedelectrically.
 25. The method of claim 24 wherein said electricallyactuating comprises interchanging the polarity of a modulation signalapplied between pairs of fixed and movable members.
 26. The method ofclaim 22 wherein said comparing reveals local asymmetry.
 27. The methodof claim 26 wherein said local asymmetry includes one of a particlebridge, vertical misalignment, variation in local etch and unequalparasitics in the interconnects between the sensors and the circuit foranalyzing.